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A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities

Eric Carlen·Elliott Lieb·Michael Loss
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Abstract

We prove a sharp analog of Young's inequality on $S^N$, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young's inequality on $R^N$ to more than three functions, and leads to significant new information about the optimizers and the constants.

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Eric Carlen, Elliott Lieb, Michael Loss. 2004-08-02. A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities. https://arxiv.org/abs/math/0408030

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